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Vibro-acoustic modelling of immersed cylindrical shells with variable thickness
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55 56 57 58 International Journal of Naval Architecture and Ocean Engineering 59 60 journal homepage: http://www.journals.elsevier.com/ 61 international-journal-of-naval-architecture-and-ocean-engineering/ 62 63 64 65 66 67 68 69 a, b, * , Hongzhou Lin c, Yue Zhu c, Weiguo Wu a, c 70 Q7,2 Xianzhong Wang 71 a Key Laboratory of High Performance Ship Technology (Wuhan University of Technology), Ministry of Education, Wuhan 430063, China b 72 Faculty of Engineering and Physical Sciences, University of Southampton, Southampton SO16 7QF, UK c School of Transportation, Wuhan University of Technology, Wuhan 430063, China 73 74 75 76 a r t i c l e i n f o a b s t r a c t 77 Article history: 78 Based on the precise transfer matrix method (PTMM), the dynamic model is constructed to observe the Received 9 July 2019 vibration behaviour of cylindrical shells with variable thickness by solving a set of first-order differential 79 Received in revised form equations. The free vibration of stiffened cylindrical shells with variable thickness can be obtained to 80 6 November 2019 compare with the exact solution and FEM results. The reliability of the present method of free vibration is 81 Accepted 19 December 2019 well proved. Furthermore, the effect of thickness on the vibration responses of the cylindrical shell is also 82 Available online xxx discussed. The acoustic response of immersed cylindrical shells is analyzed by a pluralized wave su83 perposition method (PWSM). The sound pressure coefficient can be gained by collocating points along Keywords: 84 the meridian line to satisfy the Neumann boundary condition. The mode convergence analysis of the Free vibration 85 cylindrical shell is carried out to guarantee calculation precision. Also, the reliability of the present Sound radiation 86 method on sound radiation is verified by comparing with experimental results and numerical results. Transfer matrix method 87 © 2019 Production and hosting by Elsevier B.V. on behalf of Society of Naval Architects of Korea. This is an Cylindrical shell Variable thickness 88 open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). 89 90 91 92 shell with infinite rigid baffles in fluid. Both the above models 1. Introduction 93 should be modified when the endplates are active radiators. 94 Laulagnet and Guyader (1989) explored the issues of how finite Stiffened cylindrical shells with high compressive strength are 95 cylindrical shell submerged in light and heavy fluids affected the widely introduced into engineering structures, especially in the 96 sound radiation. Under the assumption of rigid baffles on both marine field. The radiated noise of submarine which results from 97 ends, some researchers also analyze the radiated noise of cylinstructure vibration is a very important issue. Therefore, there are 98 drical shells. Up to now, there are many analysis methods including significant theoretical value and engineering meaning to study the 99 wave propagation method (Caresta and Kessissoglou, 2009), vibro-acoustic behaviour of submerged stiffened cylindrical shells. 100 modified variational method (Jin et al., 2017), transfer matrix A great number of theoretical researches have been done on the 101 method (Wang et al., 2015), Wittrick-Williams algorithm (Elvibration and radiated noise of cylindrical shells. Tottenham and 102 Kaabazi and Kennedy, 2012), extended wave method (Poultangari Shimizu (1972) put forward a transfer matrix method to analyze 103 and Nikkhahbahrami, 2016) and reverberation-ray matrix (Tang Q3 104 the free vibration characteristics of a cylindrical shell, but the method is hard to analyze the dynamic response of the shell in et al., 2017) applied to solve the vibro-acoustic problem of the cy105 fluid. Sandman (1976) investigated the acoustic loading and oblindrical shell. Nevertheless, most of the researchers have been 106 tained the generalized velocity distribution. The influence of the concentrating on cylindrical shells of equal thickness, and there are 107 baffle was observed in the axial pressure variation along the surfew applications of variable thickness shells in vibro-acoustic 108 face. After combining modal superposition with radiation impedanalysis. Obliviously, it's difficult to copy with the fluid load and 109 ance method, Stepanishen (1982) developed an approach to discontinuity of the stiffened shell. 110 evaluate the vibration and sound radiation of a finite cylindrical Considering the complexity of the governing equations of the 111 cylindrical shell, it's hard to obtain the vibration and acoustic 112 response directly of the cylindrical shell with variable thickness by 113 an analytical method. Many numerical methods such as finite * Corresponding author. Key Laboratory of High Performance Ship Technology 114 (Wuhan University of Technology), Ministry of Education, Wuhan 430063, China. element method (Petyt and Lim, 1978), boundary element method 115 E-mail address: [email protected] (X. Wang). (Ventsel et al., 2010) and coupled finite element and boundary 116 Peer review under responsibility of Society of Naval Architects of Korea. 117 118 https://doi.org/10.1016/j.ijnaoe.2019.12.003 119 2092-6782/© 2019 Production and hosting by Elsevier B.V. on behalf of Society of Naval Architects of Korea. This is an open access article under the CC BY-NC-ND license Contents lists available at ScienceDirect
Vibro-acoustic modelling of immersed cylindrical shells with variable thickness
(http://creativecommons.org/licenses/by-nc-nd/4.0/).
Please cite this article as: Wang, X et al., Vibro-acoustic modelling of immersed cylindrical shells with variable thickness, International Journal of Naval Architecture and Ocean Engineering, https://doi.org/10.1016/j.ijnaoe.2019.12.003
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element method (Liu and Chen, 2009) have potential advantages to analyze the dynamic behaviour of arbitrary complex elastic structures. However, the solution accuracy of these numerical methods relies heavily on meshed elements and frequency band. The aim of the present work is to predict the vibro-acoustic response of immersed stiffened cylindrical shell with variable thickness. After combining a precise transfer matrix method (Wang et al., 2015) and a pluralized wave superposition method (Wang et al., 2016), the authors develop an approach to observe the natural frequencies and radiated noise of cylindrical shells. The convergence analysis of modes (m, n) of the cylindrical shell is carried out to gain the excellent convergence guarantee and calculation precision. The validity and reliability of the present method for predicting free vibration are well proofed by comparing
ðNs ; Vs ; Ms Þ ¼
respectively. FðxÞ and pðxÞ are the dimensionless external force vector and sound pressure load vector. Ms , Vs , Ssq , Ns denote the moment, radial, circumferential and axial forces, respectively. The symbols with over-tilde represent the dimensionless variables. These dimensionless state vectors including four end genetic forces and four displacement functions are
ðu; w; jÞ ¼ h
1 X X
b¼0 n
1 X X ap ~ = RÞsin nq þ ap ; v ¼ h ~ ; w; ~ j ~vcos nq þ ðu 2 2 n
b¼0
(2a)
1 X 1 X X K X ap ~ ~ s; V ~ s; M ~ s RÞsin nq þ ap ; S ¼ K ð N cos n q þ S s q s q 2 2 R2 b¼0 n R2 b¼0 n
the natural frequencies with analytical results and FEM results. Furthermore, the effect of variable thickness on the free vibration of the cylindrical shell is discussed. The effectiveness of the present method for evaluating sound radiation of immersed cylindrical shells is also demonstrated by comparing the sound pressure level with experiment results and numerical results.
(2b)
where a ¼ 1 denotes the symmetric mode, a ¼ 0 denotes the antisymmetric mode. Bending rigidity K equals Eh3 =12ð1 n2 Þ, and E, n, h are Young's modulus, Poisson's ratio and thickness of the shell respectively. 2.2. The ring-stiffener
2. The dynamic model 2.1. The thin shell A diagrammatic drawing of the stiffened immersed cylindrical shell is shown in Fig. 1. h(x), L, and R denote the thickness, length and height of the cylindrical shell, respectively. A transfer matrix differential equation form of Flügge equations can be obtained
dZðxÞ = dx ¼ UðxÞZðxÞ þ FðxÞ pðxÞ
(1)
where UðxÞ is the field transfer matrix of the thin shell. The nonzero elements of UðxÞ can be referred to Wang and Guo (2016). ZðxÞ ¼ ~ M ~ ~s V ~s ~ s g. u, v, w, j denote the ~ ~v w ~ j Ssq N ffu axial, circumferential, radial displacements, angular angle,
The vibration of the stiffener could result in four kinds of forms of vibration of the shell, two of which are movements within the plane, namely flexural and extending vibration within the plane, and the other are flexural and torsional movements out of the plane. The ring-stiffeners are laid on the internal surface of the shell. If the kth stiffener and the cylindrical shell are welded together at a position xk , there are differences of internal forces between the state vector ZðxLk Þ at the left end xLk and the state vector R
R
Zðxk Þ at the right end xk . The state vector Zðxk Þ satisfies
Z xRk ¼ Trk Z xLk L
(3) R
where Zðxk Þ and Zðxk Þ denote state vectors of both ends,
Fig. 1. Scheme for the immersed stiffened cylindrical shell.
Please cite this article as: Wang, X et al., Vibro-acoustic modelling of immersed cylindrical shells with variable thickness, International Journal of Naval Architecture and Ocean Engineering, https://doi.org/10.1016/j.ijnaoe.2019.12.003
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Fig. 2. Convergence analysis of mode (m, n) on the vibro-acoustic response.
respectively. The effect of the ring-stiffener plate is considered only to change the transfer relation of state vectors at the left and right ends of the action point. Hence, Trk is obtained in a point transfer matrix form, the nonzero elements of which is given in Appendix A.
2.3. The added mass The position of the ith added mass in the shell is located at (x0i,
q0i), i ¼ 1,2, …,N, the mass of which is mi. By dealing with the ith added mass mi dðx0i Þdðq0i Þ with an orthogonal transformation, one obtains
Table 1 Comparison of natural frequencies of the cylindrical shell in vacuo (unit:Hz). n¼0
m¼1 m¼2 m¼3 m¼4 m¼5 m¼6
n¼1
Present method
Exact solution
Error
Present method
Exact solution
Error
8011.2 8205.4 8288.3 8453.6 8780.3 9341.2
8011.06 8204.57 8286.39 8450.23 8775.06 9334.10
0.00% 0.01% 0.02% 0.04% 0.06% 0.08%
4944.0 7309.7 7910.6 8256.5 8670.0 9281.6
4943.79 7308.55 7908.31 8252.68 8664.38 9274.18
0.00% 0.02% 0.03% 0.05% 0.06% 0.08%
Present method
Exact solution
Error
Present method
Exact solution
Error
2835.5 5664.3 6998.5 7737.3 8370.5 9119.3
2833.37 5661.85 6994.89 7732.17 8363.75 9110.79
0.08% 0.04% 0.05% 0.07% 0.08% 0.09%
1749.2 4243.4 5951.6 7062.8 7962.9 8898.5
1741.54 4237.93 5945.51 7055.49 7954.11 8888.28
0.44% 0.13% 0.10% 0.10% 0.11% 0.11%
n¼2
m¼1 m¼2 m¼3 m¼4 m¼5 m¼6
n¼3
Please cite this article as: Wang, X et al., Vibro-acoustic modelling of immersed cylindrical shells with variable thickness, International Journal of Naval Architecture and Ocean Engineering, https://doi.org/10.1016/j.ijnaoe.2019.12.003
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Table 2 Comparison of natural frequencies of the immersed cylindrical shell (unit:Hz).
(1,2) (1,3) (2,2) (2,3) (3,3) (1,4) (2,4) (3,4)
Present method
Wave propagation method
FEM
Error
4.94 8.91 10.86 11.56 14.68 18.24 18.68 20.14
4.92 9.06 10.71 11.24 14.7 18.68 19.14 20.37
4.95 8.95 10.66 11.54 14.73 18.26 18.71 20
0.41% 1.66% 1.40% 2.85% 0.14% 2.36% 2.40% 1.13%
Table 3 Comparison of natural frequencies of the cylindrical shell with variable thickness (unit:Hz).
Present method
FEM
Error
n¼0 n¼1 n¼2 n¼3 n¼4 n¼0 n¼1 n¼2 n¼3 n¼4 n¼0 n¼1 n¼2 n¼3 n¼4
m¼1
m¼2
m¼3
m¼4
m¼5
7971.49 5448.93 2799.47 1826.50 1679.89 7971.30 5416.20 2782.70 1803.50 1641.40 0.00% 0.60% 0.60% 1.28% 2.34%
8222.84 7326.32 5699.95 4351.26 3568.28 8164.00 7277.90 5665.50 4321.20 3531.20 0.72% 0.67% 0.61% 0.70% 1.05%
8371.82 8007.15 7135.97 6180.04 5451.98 8302.30 7944.50 7086.60 6139.30 5409.60 0.84% 0.79% 0.70% 0.66% 0.78%
8708.67 8529.87 8073.98 7518.60 7055.56 8625.60 8452.00 8007.90 7462.70 7001.30 0.96% 0.92% 0.83% 0.75% 0.78%
9367.78 9281.85 9060.99 8795.40 8594.51 9271.00 9189.00 8977.60 8720.80 8521.80 1.04% 1.01% 0.93% 0.86% 0.85%
mðxi ; qi Þ ¼ dðx0i Þ
ap mni sin nq þ 2 n¼0
1 X ∞ X
a¼0
(4)
1; n ¼ 0 where mni ¼ m0 2εnp sin nq0 þ ap . 2 , εn ¼ 2; ns0 There are differences mni u2 XðX ¼ u; v; w; jÞ between the state vector ZðxLs Þ and the state vector ZðxRs Þ at a position xs ¼ x0i =R. The state vector Zðxs Þ satisfies
Z xRs ¼ P ni Z xLs
(5)
where ZðxLk Þ and ZðxRk Þ denote state vectors at the left end xLs and the right end xRs , respectively. P ni is the point transfer matrix of ith added mass.
2.4. The exciting force A concentrated force f ðx; qÞ is introduced to act on the point (x0, q0) of the stiffener or the shell. An orthogonal transformation is carried out to deal with the formula f(x,q) ¼ f0 dðx0 Þdðq0 Þ. After P∞ ap ap multiplying the formula by and p¼0 sin pq þ 2 sin nq þ 2 integrating from 0 to 2p, one obtains
Fig. 3. Variations of natural frequencies of the cylindrical shell with various coefficient B.
Please cite this article as: Wang, X et al., Vibro-acoustic modelling of immersed cylindrical shells with variable thickness, International Journal of Naval Architecture and Ocean Engineering, https://doi.org/10.1016/j.ijnaoe.2019.12.003
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Fig. 4. Variations of natural frequencies of the cylindrical shell versus the coefficient B.
f ðx; qÞ ¼ dðx0 Þ
1 X ∞ X
fn sin nq þ
a¼0 n¼0
the relation between Zðxj Þ and Zðxjþ1 Þ can be constructed as
ap
(6)
2
1; n ¼ 0 where fn ¼ f0 2εpn sin nq0 þ ap . 2 , εn ¼ 2; ns0
ðxi
xð jþ1
2.5. The vibrational responses
Z xjþ1 ¼ e xi
Z xjþ1 ¼ exp U xjþ1 xj Z xj þ
Tri e
Z xj þ
!
UðsÞds
ðxi et
f ðtÞdt
xð jþ1
UðsÞds
xð jþ1
þ
exp U xjþ1 t rðtÞdt
xj
ðxi
xj
Case I. If there is no ring rib and added mass in the jth segment (xj ~ xjþ1 ), the relation between Zðxj Þ and Zðxjþ1 Þ can be constructed as xð jþ1
UðtÞdt
UðtÞdt
e t
f ðtÞdt
xi
xj
(8) (7)
Case II. If there is a ring rib located in the jth segment (xj ~ xjþ1 ),
Case III. If there is an added mass located in the jth segment (xj ~ xjþ1 ), the relation between Zðxj Þ and Zðxjþ1 Þ can be constructed as
Please cite this article as: Wang, X et al., Vibro-acoustic modelling of immersed cylindrical shells with variable thickness, International Journal of Naval Architecture and Ocean Engineering, https://doi.org/10.1016/j.ijnaoe.2019.12.003
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Fig. 5. Variations of natural frequencies of the cylindrical shell with various coefficient A.
whole structure can be transferred into
ðxi
xð jþ1
Z xjþ1 ¼ e xi
UðtÞdt
UðtÞdt P ni exj
ðxi Z xj þ
!
UðsÞds
ðxi et
f ðtÞdt
xj xð jþ1
T 2 6 0 6 6 0 6 4 0 0
I8 T 3 0 0 0
0 I8 T 4 0 0
0 0 I8 ::: 0
0 0 0 I8 T N
9 8 38 0 > Zðx1 Þ > > > > > > > > > > 07 7< Zðx2 Þ = < 07 ¼ x Þ Zð 3 7> > > > 0 5> « > > > > > > : ; : I8 ZðxN Þ
f ðtÞdt
e t xi
(9) The relation between Zðxj Þ and Zðxjþ1 Þ can be simplified as
Z xjþ1 ¼ T jþ1 Z xj þ P jþ1 ; j ¼ 1; :::; N 1 R xjþ1
UðtÞdt
R xjþ1
(10)
e½Uðxjþ1 tÞrðtÞ dt
where T jþ1 ¼ e , P jþ1 ¼ x in Case I; j R xi R xjþ1 UðtÞdt UðtÞdt T jþ1 ¼ e xi Tr i e xj , R xjþ1 R R xjþ1 xi Rx R UðtÞdt x P jþ1 ¼ e xi Tr i x i e t UðsÞds f ðtÞdt þ x jþ1 e t UðsÞds f ðtÞdt in j i R xi R xjþ1 UðtÞdt UðtÞdt Case II; T jþ1 ¼ e xi P ni e xj , P jþ1 ¼ R xjþ1 R xi R xjþ1 R R UðtÞdt x x e xi P ni x i e t UðsÞds f ðtÞdt þ x jþ1 e t UðsÞds f ðtÞdtin Case III. xj
j
P2 P3 P4 « PN
9 > > > = > > > ; (11)
UðsÞds
xð jþ1
þ
2
i
According to the ideas of finite element assembling mass matrix and rigidity matrix, Eq. (10) is assembled and the equations of the
Constrained stiffness constants ku , kv , kw and kq are employed to restrain displacements in the axial, tangential, radial and rotational directions at both ends of the cylindrical shell. The dimensionless internal forces and displacements at both ends satisfy the relationship
~ ¼e ~ x ¼e ~x ¼ e ~x ¼e ~ S ~ V ~; N ~ kq f; kw w; kv v ku u M xq
(12)
where e ks ¼ ks R2 h =K, s ¼ u; v; w; q are dimensionless stiffness coefficients. Classic boundary conditions including S (Simply-support), C (Clamped) and F (Free) are special cases of E (Elastic restrain) boundary conditions. When the dimensionless stiffness coefficients are generally taken as 106, the displacement result at the end tends to be zero which is approximately equivalent to the clamped boundary. The not restrained displacement is Free boundary if the dimensionless stiffness coefficient is set as 0. By identifying the line numbers of the determinate state vectors in Zðx1 Þ, ZðxN Þ, deleting the corresponding row and column in the coefficient matrix in Eq. (11) and expanding the coefficient matrix determinant, the dispersion equation of the cylindrical shell can be
Please cite this article as: Wang, X et al., Vibro-acoustic modelling of immersed cylindrical shells with variable thickness, International Journal of Naval Architecture and Ocean Engineering, https://doi.org/10.1016/j.ijnaoe.2019.12.003
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
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Fig. 6. Variations of natural frequencies of the cylindrical shell versus the coefficient A.
constructed to gain the natural frequencies of the stiffened cylindrical shell.
2.6. The acoustic responses A virtual interior surface S0 is constructed with virtual sources by shrinking the real surface S. The fluid load p is expressed as:
pðPÞ ¼ ∬ S0 sðOÞKðP; OÞdS0
(13)
respectively. sðOÞ denotes the unknown distribution function of the virtual source strength density. The source strength function K(P,O) and the distribution function sðOÞ can be expanded into a complex Fourier series. The distance d(P,O) equals qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 r P þ r O 2rP rO cosðqP qO Þ þ ðzP zO Þ . A complex distance d satisfying ð1 þigÞd is employed to avoid the non-uniqueness solution at certain characteristic frequencies. The integral interval ð0 L0 Þ and the integral interval ð0 2pÞ are divided into M and N equal divisions respectively, Eq. (13) yields
where P and O denote the external field point and source point,
Fig. 7. Comparison of sound pressure level of Model Ⅳ.
Fig. 8. Comparison of sound pressure level of Model Ⅴ.
Please cite this article as: Wang, X et al., Vibro-acoustic modelling of immersed cylindrical shells with variable thickness, International Journal of Naval Architecture and Ocean Engineering, https://doi.org/10.1016/j.ijnaoe.2019.12.003
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3. Results
pðPÞ ¼
þ∞ X
þ∞ X
~ inqP cmn Kmn ðPÞe
(14)
n¼∞ m¼∞
~ ¼ 2pL0 PM1 PN1 K½P; ~ k ; k rL eimk1 2Mp eink2 2Np is the where Kmn ðPÞ 1 2 O k1 ¼0 k2 ¼0 MN generalized sound pressure load. L0 denotes the meridian line ~ can be obtained by the length of the virtual source surface. Kmn ðPÞ discrete Fourier transform method quickly and accurately. By expanding the potential function into a complex Fourier series, the normal derivative of fluid load p is expressed as þ∞ þ∞ X X vpðPÞ 0 ~ inqP ~ OÞe ¼ cmn Kmn ðP; vnP n¼∞ m¼∞
(15)
The whole points (xi , i ¼ 1, 2, …, m) on the meridian line of real surface S should satisfy the Neumann boundary condition, which yields
K 0n c n ¼ r0 u2 wn ; n2ð∞ þ ∞Þ
(16)
where wn ¼ f wn ðx1 Þ / wn ðxm1 Þ wn ðxm Þ gT and cn ¼
T cm;n / cm1;n cmn . r0 denotes the density of the fluid media. u denotes the circular frequency. The radial displacement w satisfying the linear superposition principle, one can get
~ ¼ wf ðPÞ ~ þ wn ðPÞ n
þ∞ X
3.1. Convergence analysis The convergence of numerical results involves the intercepting mode error. The geometric parameters of a simply-supported cylindrical shell are given as follows: radius R ¼ 0.2 m, length L ¼ 0.6 m, thickness h ¼ 3 mm, size of the internal stiffener is 2 mm 30 mm, stiffener interval is 0.2 m. The radial unit force acts on the position (L/20o , 0.2 m) (cylindrical coordinate) of the internal surface. The measuring point is located at the position (L/20o , 1.2 m). The frequency domain is 40~4 kHz and the frequency step is 40 Hz. The shell and stiffener are made of steel, with density r ¼ 7850 kg/m3, Poisson's ratio m ¼ 0.3, Young's modulus of elasticity E ¼ 2.06 1011 Pa, and damping loss factor h ¼ 0.01. The sound speed of the fluid c0 ¼ 1500 m/s and the density r0 ¼ 1000 kg/m3. As shown in Fig. 2, a convergence is brought about with few modes in low frequency range. More modes are employed to ensure high accuracy as the frequency increases. The number of axial wavenumber m and circumferential wavenumber n take 0e5 and 1e35 respectively in order to realize the accuracy guarantee in the calculated frequency range.
3.2. The free vibration a) Comparison: Model I
p ~ cmn wmn ðPÞ
(17)
m¼∞
~ denotes the radial displacement under the external where wfn ðPÞ load FðxÞ. wpmn ðxÞ denotes the radial displacement under generalized sound pressure load Kmn . ~ on the meridian line of the surface S After collocating points P and substituting Eq. (17) into Eq. (16), one can get
U n c n ¼ Q n ; n2ð∞ þ ∞Þ
(18)
0 ðPÞ ~ r u2 wp ðPÞ, ~ ~ where U q2mþ1 ¼ Kmn Q q1 ¼ r0 u2 wfn ðPÞ. mn 0 Moore-Penrose generalized inverse method is employed to solve the unknown coefficient vector c n. Then the radiated noise of immersed cylindrical shells is obtained by the pluralized wave superposition method (PWSM).
In order to prove the effectiveness of the present method, the geometric and physical parameters of the cylindrical shell (Model I) with simply supported boundary conditions are given as follows: radius R ¼ 0.1 m, length L ¼ 0.2 m, thickness h ¼ 2 mm, density r ¼ 7850 kg/m3, Poisson's ratio m ¼ 0.3 and Young's modulus of elasticity E ¼ 2.1 1011 Pa. The natural frequencies for each order (m, n) are calculated and compared with the exact solution as shown in Table 1. The natural frequencies obtained by the present method are basically consistent with the exact solution. The maximum relative errors are no more than 0.5%, which demonstrates the reliability of the present method on free vibration of the cylindrical shell. b) Comparison: Model II The emphasis of this part is to fully verify the effectiveness of the present method on natural frequencies of the submerged cylindrical shell. The geometric and physical parameters of the model II (Zhang, 2002) are given as follows: radius R ¼ 1.0 m, length L ¼ 20 m, thickness h ¼ 10 mm, Poisson's ratio m ¼ 0.3, Young's modulus of elasticity E ¼ 2.1 1011 Pa and density r ¼ 7850 kg/m3. Sound speed c0 in water equals 1500 m/s. The density of water r0 ¼ 1000 kg/m3. For a clamped cylindrical shell, the natural frequencies of the immersed shell are compared with analytical results and FEM result, as shown in Table 2. The maximum relative error is no more than 3%, which shows that the present method is consistent with FEM results and wave propagation method. It also demonstrates the reliability of the present method on free vibration of the immersed cylindrical shell. c) Comparison: Model III
Fig. 9. Comparison of sound pressure level of Model Ⅵ.
Model III is constructed to validate the reliability of the method on free vibration of the cylindrical shell with variable thickness. The geometric and physical parameters of the Model III are given as follows: radius R ¼ 0.1 m, length L ¼ 0.2 m, Poisson's ratio m ¼ 0.3,
Please cite this article as: Wang, X et al., Vibro-acoustic modelling of immersed cylindrical shells with variable thickness, International Journal of Naval Architecture and Ocean Engineering, https://doi.org/10.1016/j.ijnaoe.2019.12.003
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Young's modulus of elasticity E ¼ 2.1 1011 Pa and density
r ¼ 7800 kg/m3. The thickness h equals h0 ð1 þ Ax=LÞB , which is the function of the axial coordinate value x. Both A and B are suggested to be 1. h0 is set to 2 mm. The natural frequencies of the simply supported cylindrical shell with variable thickness are compared with FEM results, as shown in Table 3. From Table 3, the natural frequencies obtained by the present method are in good agreement with FEM results. The maximum relative difference is no more than 3%. To keep both the thickness of both ends to be constant (the thickness of left end hl ¼ 2 mm and the thickness of right end hr ¼ 4 mm), the model III was employed to analyze the influence of thickness coefficient A and B on free vibration of the cylindrical shell with various thickness. When circumferential wavenumber n is 2, 4, 6 and 8 respectively, the relation between natural frequencies and thickness coefficient B was analyzed, as shown in Fig. 3 and Fig. 4. The effect of the thickness coefficient B can be ignored when circumferential wavenumber n is less than 4. The difference becomes more obvious as the circumferential wavenumber n and axial wavenumber m increase. For a given wavenumber n, the curve shows the antisymmetric characteristic. When the thickness coefficient B is negative, the natural frequency goes up as the thickness coefficient B increases. It comes to an opposite conclusion when the thickness coefficient B is positive. When circumferential wavenumber n was 2, 4, 6 and 8 respectively, the relation between natural frequencies and thickness coefficient A was also analyzed, as shown in Fig. 5 and Fig. 6. The effect of the thickness coefficient A is in accordance with that of the thickness coefficient B when circumferential wavenumber n is less than 4. The only difference is that the coefficient A affects the natural frequency when m ¼ 1 and n ¼ 1. The difference of natural frequencies become more obvious as the circumferential wavenumber n and axial wavenumber m increase. The natural frequency goes up as the thickness coefficient A increases when n ¼ 4, 6 and 8. When n ¼ 2 the curve is approximately a horizontal line, which means thickness coefficient A has little impact on the natural frequency.
3.3. The acoustic response a) Comparison analysis: Model Ⅳ
Fig. 10. Comparison of sound pressure level of Model Ⅶ: (a) hydrophone 1, (b) hydrophone 2, (c) hydrophone 3.
The geometric parameters of the Model Ⅳ are given as follows: radius R ¼ 0.175 m, length L ¼ 0.6 m, thickness h ¼ 2 mm. The centre of the left end of the cylindrical shell was defined cylindrical coordinate origin. The radial unit force acts on the point (L/ 2,0 ,0.175) (cylindrical coordinates) of the cylindrical shell with excitation frequency f ¼ 4 kHz. The hydrophone is fixed at the position of 1 m from the external surface of the shell. The physical parameters of the shell are given as follows: Poisson's ratio m ¼ 0.3, Young's modulus of elasticity E ¼ 2.1 1011 Pa, density r ¼ 7850 kg/m3, and loss factor h ¼ 0.01. Sound speed c0 in water equals 1500 m/s. The density of water r0 ¼ 1000 kg/m3. The simply supported boundary conditions at both ends are assumed. The results contrast of sound pressure level Lp is yielded by adopting the present method, experimental method and B. Laulgnet's method respectively, as shown in Fig. 7. The error curve shows the maximum error of the sound pressure level between the present method and experiment results. It can be seen from Fig. 7 that the error of sound pressure level is 0.5%e5% at other angles except at 60 , which accords with the allowable range of error. It proves that the present method is feasible, effective and sufficiently accurate and can be applied to predict sound radiation of cylindrical shells.
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b) Comparison analysis: Model Ⅴ The geometric parameters of the Model Ⅴ with simply supported boundary condition are given as follows: radius R ¼ 0.2 m, length L ¼ 0.6 m, thickness h ¼ 3 mm. Sectional dimensions of the stiffener is 2 mm 30 mm. The stiffeners spacing Dl ¼ 0.2 m. The radial unit force acts on the point (0.3 m, 0 , 0.2 m) (cylindrical coordinates) of the shell with excitation frequency f ¼ 4 kHz. The hydrophone is fixed at the position of 1 m from the external surface of the shell. The physical parameters are the same as model Ⅳ. It can be seen from Fig. 8 that the relative error of sound pressure level between the present method and wave propagation method is no more than 6%. The results from the present method also are consistent with the experimental values, except for some minor differences in some partial angle. Hence, the present method proposed can be applied to vibro-acoustic behaviour of the stiffened cylindrical shell. c) Comparison analysis: Model Ⅵ Model Ⅵ with the variable thickness is constructed to validate the availability and accuracy of the present method on the sound radiation of the cylindrical shell with the variable thickness. The geometric and physical parameters of the Model Ⅵ with clamped boundary condition are given as follows: the variation coefficient A ¼ 1, the power coefficient B ¼ 1. The other geometric and physical parameters are the same as model Ⅴ, except without ring ribs. The radial unit force acts with excitation frequency range 10 Hz-4kHz. The observation point located at position (0.3 m, 0 , 1.2 m), in which acoustic responses of Model Ⅵ are calculated by coupled FEM/BEM. C3D8R elements in software ABAQUS are employed to generate linear hexahedron meshes of the cylindrical shell, the size of which is set to 4 mm. The external flow field radius is taken as 2 m. A hemispherical flow field is meshed by AC3D4 acoustic tetrahedral elements, the size of which is from 0.004 m to 0.0125 m. The results comparison of radiated noises at the observation point between the present method and coupled FEM/BEM is shown in Fig. 9. The results from the present method are consistent with the results from coupled FEM/BEM at 10 Hze2100 Hz. Although the sound pressure level is slightly different at the resonance peak value in the 2100~4 kHz range, the sound pressure values resulted from the present method is in accord with numerical results basically. There are many reasons for the difference. The process of equal divisions along the integral interval in Eq. (14) could cause the error of approximation. Both the substitution of the finite field for infinite fluid field and the setting of Tie connection at the interface also lead to the numerical error in software ABAQUS. The comparison provides the validity and reliability of the present method, which can be applied to sound radiation of immersed cylindrical shells with the variable thickness. d) Comparison analysis: Model Ⅶ The test model of a ring-stiffened cylindrical shell is constructed and prepared to prove the effectiveness of the present method. Model Ⅶ has the following material properties: Length L ¼ 0.8 m, Radius R ¼ 0.3 m. Shell thickness t ¼ 4 mm. Sectional dimensions of the stiffener is 40 mm 4 mm. The stiffeners spacing Dl ¼ 0.16 m. The other physical parameters are the same as the Model Ⅳ. 15 mm thick caps are screwed at the ends of the cylinder. The shaker is fixed on the end cap in order to excite the point (0.4 m,0 ,0.3 m) of Model Ⅶ. There are 3 hydrophones (BK-8104) is placed 1.0 m far from the cylindrical shell, which was set up in the anechoic tank to measure the sound pressure. Comparison of sound radiation curves between model test and the present method with the linear sweep
frequency (100 Hz-2kHz) is given in Fig. 10. It can be observed that although the sound pressure is slightly different at the resonance peak value in the 500e2000 Hz band, the sound pressure result from PTMM basically is in accord with test result. When the internal medium of the experimental model is full of air, the coupling between structural mode and acoustic cavity mode has little effect on the vibro-acoustic response of the cylindrical shell. In general the present method provides enough verification and accuracy, which can be applied to vibro-acoustic behaviours of immersed stiffened cylindrical shells. 4. Conclusions The vibro-acoustic model of the immersed stiffened cylindrical shell with various thickness is developed in this paper to predict the vibro-acoustic performance by combining PTMM and PWSM. Based on thin shell theory, a global matrix formulation as the governing equation is constructed with a set of field matrixes of stiffened cylindrical shell segments with various thickness. Then the dynamic behaviours can be directly solved by PTMM. The fluid load simulated by PWSM is employed to evaluate the radiated noise of the immersed cylindrical shell. Free vibration of three different cylindrical shells including cylindrical shell, immersed cylindrical shell and variable-thickness cylindrical shell are constructed and analyzed to illustrates the feasibility of the present method. Acoustic responses of four different cylindrical shells including cylindrical shell, immersed cylindrical shell, stiffened cylindrical shell and variable-thickness cylindrical shell are also constructed and analyzed to demonstrate its effectiveness. It is worth to point out that the proposed method is conveniently extended to analyze the vibro-acoustic behaviours of other elastic shells like conical shells, double-walled shells and combined shells with various thickness. Declaration of competing interest We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled, “Vibro-acoustic modelling of immersed cylindrical shells with variable thickness”. Acknowledgements The authors would like to thank the anonymous reviewers for their valuable comments. This paper was financially supported by China Scholarship Council (201806955052), National Natural Science Foundation of China (51779201, 51609190) and Natural Sci- Q5 ence Foundation of Hubei Province (2018CFB607). Appendix A
Trði; iÞ ¼ 1; i ¼ 1; 2; 3/8 R Trð5; 1Þ ¼ K
("
EI2 R3b
2
n þ
GJ R3b
# 2
n
h þ eh
" EI2 R4b
4
n þ
GJ R4b
#) n rAu 2
2
Please cite this article as: Wang, X et al., Vibro-acoustic modelling of immersed cylindrical shells with variable thickness, International Journal of Naval Architecture and Ocean Engineering, https://doi.org/10.1016/j.ijnaoe.2019.12.003
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Trð5; 4Þ ¼ # " 8 9 EI2 EI2 2 GJ 2 GJ 2 2 > > > > R en en þ R n r I u p > > 3 b 3 3 3 b > > < = R R R R b b b b h " # > > K> > > > > : þe EI2 en4 þ EI2 Rn2 GJ en2 þ GJ Rn2 rAu2 e > ; R4b R4b R4b R4b " # R2 EI1 Rb 3 EA Rb n þ 2 Trð6; 2Þ ¼ h K R4 R Rb R b Trð6; 3Þ ¼ h
" # R2 EI1 Rb 4 EA e 2 2 n n þ r A u 1 þ R K R4 R R2b b
Trð7; 2Þ ¼ h
" # R2 EI1 Rb 2 EA Rb 2 2 Rb n n þ r A u K R4 R R R2b R b
Trð7; 3Þ ¼ h
# " R2 EI1 Rb 3 EA R e 2 e n þ n r A u n R R K R4 R R2b b
Trð8; 1Þ ¼ h
# " R2 EI2 4 GJ 2 2 n þ n r A u K R4 R4b b
# " h R2 EI2 4 EI2 2 GJ 2 GJ 2 2 Trð8; 4Þ ¼ 4 en þ 4 R1 n 4 en þ 4 R1 n rAu e R K Rb Rb Rb Rb where Rb ¼ R þ e e is the offset of the rib. E, G are the Modulus of elasticity and shear of the rib. r is the density of the rib. A is the sectional area of the rib. I1 , I2 and J are the moment of inertia in xdirection, y-direction and polar moment of inertia. Q6
Uncited references Chen et al., 2018; Duan and Koh, 2008; Irie et al., 1984; Li et al., 2019; Meyer et al., 2016; Qu et al., 2013; Senjanovic et al., 2018; Trefethen, 2000; Xie et al., 2017; Zhang and Xiang, 2007. References Caresta, M., Kessissoglou, N.J., 2009. Structural and acoustic responses of a fluid-
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loaded cylindrical hull with structural discontinuities. Appl. Acoust. 70 (7), 954e963. Chen, M., Zhang, L., Xie, K., 2018. Vibration analysis of a cylindrical shell coupled with interior structures using a hybrid analytical-numerical approach. Ocean Eng. 154, 81e93. Duan, W., Koh, C., 2008. Axisymmetric transverse vibrations of circular cylindrical shells with variable thickness. J. Sound Vib. 317 (3e5), 1035e1041. El-Kaabazi, N., Kennedy, D., 2012. Calculation of natural frequencies and vibration modes of variable thickness cylindrical shells using the WittrickeWilliams algorithm. Comput. Struct. 104, 4e12. Irie, T., Yamada, G., Muramoto, Y., 1984. Free vibration of joined conical-cylindrical shells. J. Sound Vib. 95 (1), 31e39. Jin, G., Ma, X., Liu, Z., Xuan, L., 2017. Dynamic analysis of general rotationally symmetric built-up structures using a modified fourier spectral element approach. J. Vib. Acoust. 139 (2), 021012. Laulagnet, B., Guyader, J., 1989. Modal analysis of a shell's acoustic radiation in light and heavy fluids. J. Sound Vib. 131 (3), 397e415. Li, H., Pang, F., Miao, X., Li, Y., 2019. JacobieRitz method for free vibration analysis of uniform and stepped circular cylindrical shells with arbitrary boundary conditions: a unified formulation. Comput. Math. Appl. 77 (2), 427e440. Liu, C.-H., Chen, P.-T., 2009. Numerical analysis of immersed finite cylindrical shells using a coupled BEM/FEM and spatial spectrum approach. Appl. Acoust. 70 (2), 256e266. Meyer, V., Maxit, L., Guyader, J.-L., Leissing, T., 2016. Prediction of the vibroacoustic behavior of a submerged shell with non-axisymmetric internal substructures by a condensed transfer function method. J. Sound Vib. 360, 260e276. Petyt, M., Lim, S., 1978. Finite element analysis of the noise inside a mechanically excited cylinder. Int. J. Numer. Methods Eng. 13 (1), 109e122. Qu, Y., Chen, Y., Long, X., Hua, H., Meng, G., 2013. Free and forced vibration analysis of uniform and stepped circular cylindrical shells using a domain decomposition method. Appl. Acoust. 74 (3), 425e439. Sandman, B., 1976. Fluid-loading influence coefficients for a finite cylindrical shell. J. Acoust. Soc. Am. 60 (6), 1256e1264. Senjanovi c, I., Catipovi c, I., Alujevi c, N., Vladimir, N., Cakmak, D., 2018. A finite strip for the vibration analysis of rotating cylindrical shells. Thin-Walled Struct. 122, 158e172. Stepanishen, P.R., 1982. Modal coupling in the vibration of fluid-loaded cylindrical shells. J. Acoust. Soc. Am. 71 (4), 813e823. Tang, D., Yao, X., Wu, G., Peng, Y., 2017. Free and forced vibration analysis of multistepped circular cylindrical shells with arbitrary boundary conditions by the method of reverberation-ray matrix. Thin-Walled Struct. 116, 154e168. Tottenham, H., Shimizu, K., 1972. Analysis of the free vibration of cantilever cylindrical thin elastic shells by the matrix progression method. Int. J. Mech. Sci. 14 (5), 293e310. Trefethen, L.N., 2000. Spectral Methods in MATLAB. Siam. Ventsel, E., Naumenko, V., Strelnikova, E., Yeseleva, E., 2010. Free vibrations of shells of revolution filled with a fluid. Eng. Anal. Bound. Elem. 34 (10), 856e862. Wang, X., Guo, W., 2016. Dynamic modeling and vibration characteristics analysis of submerged stiffened combined shells. Ocean Eng. 127, 226e235. Wang, X., Wu, W., Yao, X., 2015. Structural and acoustic response of a finite stiffened conical shell. Acta Mech. Solida Sin. 28 (2), 200e209. Wang, X.-z., Jiang, C.-b., Xu, R.-y., 2016. Structural and acoustic response of a finite stiffened submarine hull. China Ocean Eng. 30 (6), 898e915. Xie, K., Chen, M., Zhang, L., Xie, D., 2017. Free and forced vibration analysis of nonuniformly supported cylindrical shells through wave based method. Int. J. Mech. Sci. 128, 512e526. Zhang, X., 2002. Frequency analysis of submerged cylindrical shells with the wave propagation approach. Int. J. Mech. Sci. 44 (7), 1259e1273. Zhang, L., Xiang, Y., 2007. Exact solutions for vibration of stepped circular cylindrical shells. J. Sound Vib. 299 (4e5), 948e964.
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55 56 57 58 International Journal of Naval Architecture and Ocean Engineering 59 60 journal homepage: http://www.journals.elsevier.com/ 61 international-journal-of-naval-architecture-and-ocean-engineering/ 62 63 64 65 66 67 68 69 a, b, * , Hongzhou Lin c, Yue Zhu c, Weiguo Wu a, c 70 Q7,2 Xianzhong Wang 71 a Key Laboratory of High Performance Ship Technology (Wuhan University of Technology), Ministry of Education, Wuhan 430063, China b 72 Faculty of Engineering and Physical Sciences, University of Southampton, Southampton SO16 7QF, UK c School of Transportation, Wuhan University of Technology, Wuhan 430063, China 73 74 75 76 a r t i c l e i n f o a b s t r a c t 77 Article history: 78 Based on the precise transfer matrix method (PTMM), the dynamic model is constructed to observe the Received 9 July 2019 vibration behaviour of cylindrical shells with variable thickness by solving a set of first-order differential 79 Received in revised form equations. The free vibration of stiffened cylindrical shells with variable thickness can be obtained to 80 6 November 2019 compare with the exact solution and FEM results. The reliability of the present method of free vibration is 81 Accepted 19 December 2019 well proved. Furthermore, the effect of thickness on the vibration responses of the cylindrical shell is also 82 Available online xxx discussed. The acoustic response of immersed cylindrical shells is analyzed by a pluralized wave su83 perposition method (PWSM). The sound pressure coefficient can be gained by collocating points along Keywords: 84 the meridian line to satisfy the Neumann boundary condition. The mode convergence analysis of the Free vibration 85 cylindrical shell is carried out to guarantee calculation precision. Also, the reliability of the present Sound radiation 86 method on sound radiation is verified by comparing with experimental results and numerical results. Transfer matrix method 87 © 2019 Production and hosting by Elsevier B.V. on behalf of Society of Naval Architects of Korea. This is an Cylindrical shell Variable thickness 88 open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). 89 90 91 92 shell with infinite rigid baffles in fluid. Both the above models 1. Introduction 93 should be modified when the endplates are active radiators. 94 Laulagnet and Guyader (1989) explored the issues of how finite Stiffened cylindrical shells with high compressive strength are 95 cylindrical shell submerged in light and heavy fluids affected the widely introduced into engineering structures, especially in the 96 sound radiation. Under the assumption of rigid baffles on both marine field. The radiated noise of submarine which results from 97 ends, some researchers also analyze the radiated noise of cylinstructure vibration is a very important issue. Therefore, there are 98 drical shells. Up to now, there are many analysis methods including significant theoretical value and engineering meaning to study the 99 wave propagation method (Caresta and Kessissoglou, 2009), vibro-acoustic behaviour of submerged stiffened cylindrical shells. 100 modified variational method (Jin et al., 2017), transfer matrix A great number of theoretical researches have been done on the 101 method (Wang et al., 2015), Wittrick-Williams algorithm (Elvibration and radiated noise of cylindrical shells. Tottenham and 102 Kaabazi and Kennedy, 2012), extended wave method (Poultangari Shimizu (1972) put forward a transfer matrix method to analyze 103 and Nikkhahbahrami, 2016) and reverberation-ray matrix (Tang Q3 104 the free vibration characteristics of a cylindrical shell, but the method is hard to analyze the dynamic response of the shell in et al., 2017) applied to solve the vibro-acoustic problem of the cy105 fluid. Sandman (1976) investigated the acoustic loading and oblindrical shell. Nevertheless, most of the researchers have been 106 tained the generalized velocity distribution. The influence of the concentrating on cylindrical shells of equal thickness, and there are 107 baffle was observed in the axial pressure variation along the surfew applications of variable thickness shells in vibro-acoustic 108 face. After combining modal superposition with radiation impedanalysis. Obliviously, it's difficult to copy with the fluid load and 109 ance method, Stepanishen (1982) developed an approach to discontinuity of the stiffened shell. 110 evaluate the vibration and sound radiation of a finite cylindrical Considering the complexity of the governing equations of the 111 cylindrical shell, it's hard to obtain the vibration and acoustic 112 response directly of the cylindrical shell with variable thickness by 113 an analytical method. Many numerical methods such as finite * Corresponding author. Key Laboratory of High Performance Ship Technology 114 (Wuhan University of Technology), Ministry of Education, Wuhan 430063, China. element method (Petyt and Lim, 1978), boundary element method 115 E-mail address: [email protected] (X. Wang). (Ventsel et al., 2010) and coupled finite element and boundary 116 Peer review under responsibility of Society of Naval Architects of Korea. 117 118 https://doi.org/10.1016/j.ijnaoe.2019.12.003 119 2092-6782/© 2019 Production and hosting by Elsevier B.V. on behalf of Society of Naval Architects of Korea. This is an open access article under the CC BY-NC-ND license Contents lists available at ScienceDirect
Vibro-acoustic modelling of immersed cylindrical shells with variable thickness
(http://creativecommons.org/licenses/by-nc-nd/4.0/).
Please cite this article as: Wang, X et al., Vibro-acoustic modelling of immersed cylindrical shells with variable thickness, International Journal of Naval Architecture and Ocean Engineering, https://doi.org/10.1016/j.ijnaoe.2019.12.003
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element method (Liu and Chen, 2009) have potential advantages to analyze the dynamic behaviour of arbitrary complex elastic structures. However, the solution accuracy of these numerical methods relies heavily on meshed elements and frequency band. The aim of the present work is to predict the vibro-acoustic response of immersed stiffened cylindrical shell with variable thickness. After combining a precise transfer matrix method (Wang et al., 2015) and a pluralized wave superposition method (Wang et al., 2016), the authors develop an approach to observe the natural frequencies and radiated noise of cylindrical shells. The convergence analysis of modes (m, n) of the cylindrical shell is carried out to gain the excellent convergence guarantee and calculation precision. The validity and reliability of the present method for predicting free vibration are well proofed by comparing
ðNs ; Vs ; Ms Þ ¼
respectively. FðxÞ and pðxÞ are the dimensionless external force vector and sound pressure load vector. Ms , Vs , Ssq , Ns denote the moment, radial, circumferential and axial forces, respectively. The symbols with over-tilde represent the dimensionless variables. These dimensionless state vectors including four end genetic forces and four displacement functions are
ðu; w; jÞ ¼ h
1 X X
b¼0 n
1 X X ap ~ = RÞsin nq þ ap ; v ¼ h ~ ; w; ~ j ~vcos nq þ ðu 2 2 n
b¼0
(2a)
1 X 1 X X K X ap ~ ~ s; V ~ s; M ~ s RÞsin nq þ ap ; S ¼ K ð N cos n q þ S s q s q 2 2 R2 b¼0 n R2 b¼0 n
the natural frequencies with analytical results and FEM results. Furthermore, the effect of variable thickness on the free vibration of the cylindrical shell is discussed. The effectiveness of the present method for evaluating sound radiation of immersed cylindrical shells is also demonstrated by comparing the sound pressure level with experiment results and numerical results.
(2b)
where a ¼ 1 denotes the symmetric mode, a ¼ 0 denotes the antisymmetric mode. Bending rigidity K equals Eh3 =12ð1 n2 Þ, and E, n, h are Young's modulus, Poisson's ratio and thickness of the shell respectively. 2.2. The ring-stiffener
2. The dynamic model 2.1. The thin shell A diagrammatic drawing of the stiffened immersed cylindrical shell is shown in Fig. 1. h(x), L, and R denote the thickness, length and height of the cylindrical shell, respectively. A transfer matrix differential equation form of Flügge equations can be obtained
dZðxÞ = dx ¼ UðxÞZðxÞ þ FðxÞ pðxÞ
(1)
where UðxÞ is the field transfer matrix of the thin shell. The nonzero elements of UðxÞ can be referred to Wang and Guo (2016). ZðxÞ ¼ ~ M ~ ~s V ~s ~ s g. u, v, w, j denote the ~ ~v w ~ j Ssq N ffu axial, circumferential, radial displacements, angular angle,
The vibration of the stiffener could result in four kinds of forms of vibration of the shell, two of which are movements within the plane, namely flexural and extending vibration within the plane, and the other are flexural and torsional movements out of the plane. The ring-stiffeners are laid on the internal surface of the shell. If the kth stiffener and the cylindrical shell are welded together at a position xk , there are differences of internal forces between the state vector ZðxLk Þ at the left end xLk and the state vector R
R
Zðxk Þ at the right end xk . The state vector Zðxk Þ satisfies
Z xRk ¼ Trk Z xLk L
(3) R
where Zðxk Þ and Zðxk Þ denote state vectors of both ends,
Fig. 1. Scheme for the immersed stiffened cylindrical shell.
Please cite this article as: Wang, X et al., Vibro-acoustic modelling of immersed cylindrical shells with variable thickness, International Journal of Naval Architecture and Ocean Engineering, https://doi.org/10.1016/j.ijnaoe.2019.12.003
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3
Fig. 2. Convergence analysis of mode (m, n) on the vibro-acoustic response.
respectively. The effect of the ring-stiffener plate is considered only to change the transfer relation of state vectors at the left and right ends of the action point. Hence, Trk is obtained in a point transfer matrix form, the nonzero elements of which is given in Appendix A.
2.3. The added mass The position of the ith added mass in the shell is located at (x0i,
q0i), i ¼ 1,2, …,N, the mass of which is mi. By dealing with the ith added mass mi dðx0i Þdðq0i Þ with an orthogonal transformation, one obtains
Table 1 Comparison of natural frequencies of the cylindrical shell in vacuo (unit:Hz). n¼0
m¼1 m¼2 m¼3 m¼4 m¼5 m¼6
n¼1
Present method
Exact solution
Error
Present method
Exact solution
Error
8011.2 8205.4 8288.3 8453.6 8780.3 9341.2
8011.06 8204.57 8286.39 8450.23 8775.06 9334.10
0.00% 0.01% 0.02% 0.04% 0.06% 0.08%
4944.0 7309.7 7910.6 8256.5 8670.0 9281.6
4943.79 7308.55 7908.31 8252.68 8664.38 9274.18
0.00% 0.02% 0.03% 0.05% 0.06% 0.08%
Present method
Exact solution
Error
Present method
Exact solution
Error
2835.5 5664.3 6998.5 7737.3 8370.5 9119.3
2833.37 5661.85 6994.89 7732.17 8363.75 9110.79
0.08% 0.04% 0.05% 0.07% 0.08% 0.09%
1749.2 4243.4 5951.6 7062.8 7962.9 8898.5
1741.54 4237.93 5945.51 7055.49 7954.11 8888.28
0.44% 0.13% 0.10% 0.10% 0.11% 0.11%
n¼2
m¼1 m¼2 m¼3 m¼4 m¼5 m¼6
n¼3
Please cite this article as: Wang, X et al., Vibro-acoustic modelling of immersed cylindrical shells with variable thickness, International Journal of Naval Architecture and Ocean Engineering, https://doi.org/10.1016/j.ijnaoe.2019.12.003
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Table 2 Comparison of natural frequencies of the immersed cylindrical shell (unit:Hz).
(1,2) (1,3) (2,2) (2,3) (3,3) (1,4) (2,4) (3,4)
Present method
Wave propagation method
FEM
Error
4.94 8.91 10.86 11.56 14.68 18.24 18.68 20.14
4.92 9.06 10.71 11.24 14.7 18.68 19.14 20.37
4.95 8.95 10.66 11.54 14.73 18.26 18.71 20
0.41% 1.66% 1.40% 2.85% 0.14% 2.36% 2.40% 1.13%
Table 3 Comparison of natural frequencies of the cylindrical shell with variable thickness (unit:Hz).
Present method
FEM
Error
n¼0 n¼1 n¼2 n¼3 n¼4 n¼0 n¼1 n¼2 n¼3 n¼4 n¼0 n¼1 n¼2 n¼3 n¼4
m¼1
m¼2
m¼3
m¼4
m¼5
7971.49 5448.93 2799.47 1826.50 1679.89 7971.30 5416.20 2782.70 1803.50 1641.40 0.00% 0.60% 0.60% 1.28% 2.34%
8222.84 7326.32 5699.95 4351.26 3568.28 8164.00 7277.90 5665.50 4321.20 3531.20 0.72% 0.67% 0.61% 0.70% 1.05%
8371.82 8007.15 7135.97 6180.04 5451.98 8302.30 7944.50 7086.60 6139.30 5409.60 0.84% 0.79% 0.70% 0.66% 0.78%
8708.67 8529.87 8073.98 7518.60 7055.56 8625.60 8452.00 8007.90 7462.70 7001.30 0.96% 0.92% 0.83% 0.75% 0.78%
9367.78 9281.85 9060.99 8795.40 8594.51 9271.00 9189.00 8977.60 8720.80 8521.80 1.04% 1.01% 0.93% 0.86% 0.85%
mðxi ; qi Þ ¼ dðx0i Þ
ap mni sin nq þ 2 n¼0
1 X ∞ X
a¼0
(4)
1; n ¼ 0 where mni ¼ m0 2εnp sin nq0 þ ap . 2 , εn ¼ 2; ns0 There are differences mni u2 XðX ¼ u; v; w; jÞ between the state vector ZðxLs Þ and the state vector ZðxRs Þ at a position xs ¼ x0i =R. The state vector Zðxs Þ satisfies
Z xRs ¼ P ni Z xLs
(5)
where ZðxLk Þ and ZðxRk Þ denote state vectors at the left end xLs and the right end xRs , respectively. P ni is the point transfer matrix of ith added mass.
2.4. The exciting force A concentrated force f ðx; qÞ is introduced to act on the point (x0, q0) of the stiffener or the shell. An orthogonal transformation is carried out to deal with the formula f(x,q) ¼ f0 dðx0 Þdðq0 Þ. After P∞ ap ap multiplying the formula by and p¼0 sin pq þ 2 sin nq þ 2 integrating from 0 to 2p, one obtains
Fig. 3. Variations of natural frequencies of the cylindrical shell with various coefficient B.
Please cite this article as: Wang, X et al., Vibro-acoustic modelling of immersed cylindrical shells with variable thickness, International Journal of Naval Architecture and Ocean Engineering, https://doi.org/10.1016/j.ijnaoe.2019.12.003
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Fig. 4. Variations of natural frequencies of the cylindrical shell versus the coefficient B.
f ðx; qÞ ¼ dðx0 Þ
1 X ∞ X
fn sin nq þ
a¼0 n¼0
the relation between Zðxj Þ and Zðxjþ1 Þ can be constructed as
ap
(6)
2
1; n ¼ 0 where fn ¼ f0 2εpn sin nq0 þ ap . 2 , εn ¼ 2; ns0
ðxi
xð jþ1
2.5. The vibrational responses
Z xjþ1 ¼ e xi
Z xjþ1 ¼ exp U xjþ1 xj Z xj þ
Tri e
Z xj þ
!
UðsÞds
ðxi et
f ðtÞdt
xð jþ1
UðsÞds
xð jþ1
þ
exp U xjþ1 t rðtÞdt
xj
ðxi
xj
Case I. If there is no ring rib and added mass in the jth segment (xj ~ xjþ1 ), the relation between Zðxj Þ and Zðxjþ1 Þ can be constructed as xð jþ1
UðtÞdt
UðtÞdt
e t
f ðtÞdt
xi
xj
(8) (7)
Case II. If there is a ring rib located in the jth segment (xj ~ xjþ1 ),
Case III. If there is an added mass located in the jth segment (xj ~ xjþ1 ), the relation between Zðxj Þ and Zðxjþ1 Þ can be constructed as
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Fig. 5. Variations of natural frequencies of the cylindrical shell with various coefficient A.
whole structure can be transferred into
ðxi
xð jþ1
Z xjþ1 ¼ e xi
UðtÞdt
UðtÞdt P ni exj
ðxi Z xj þ
!
UðsÞds
ðxi et
f ðtÞdt
xj xð jþ1
T 2 6 0 6 6 0 6 4 0 0
I8 T 3 0 0 0
0 I8 T 4 0 0
0 0 I8 ::: 0
0 0 0 I8 T N
9 8 38 0 > Zðx1 Þ > > > > > > > > > > 07 7< Zðx2 Þ = < 07 ¼ x Þ Zð 3 7> > > > 0 5> « > > > > > > : ; : I8 ZðxN Þ
f ðtÞdt
e t xi
(9) The relation between Zðxj Þ and Zðxjþ1 Þ can be simplified as
Z xjþ1 ¼ T jþ1 Z xj þ P jþ1 ; j ¼ 1; :::; N 1 R xjþ1
UðtÞdt
R xjþ1
(10)
e½Uðxjþ1 tÞrðtÞ dt
where T jþ1 ¼ e , P jþ1 ¼ x in Case I; j R xi R xjþ1 UðtÞdt UðtÞdt T jþ1 ¼ e xi Tr i e xj , R xjþ1 R R xjþ1 xi Rx R UðtÞdt x P jþ1 ¼ e xi Tr i x i e t UðsÞds f ðtÞdt þ x jþ1 e t UðsÞds f ðtÞdt in j i R xi R xjþ1 UðtÞdt UðtÞdt Case II; T jþ1 ¼ e xi P ni e xj , P jþ1 ¼ R xjþ1 R xi R xjþ1 R R UðtÞdt x x e xi P ni x i e t UðsÞds f ðtÞdt þ x jþ1 e t UðsÞds f ðtÞdtin Case III. xj
j
P2 P3 P4 « PN
9 > > > = > > > ; (11)
UðsÞds
xð jþ1
þ
2
i
According to the ideas of finite element assembling mass matrix and rigidity matrix, Eq. (10) is assembled and the equations of the
Constrained stiffness constants ku , kv , kw and kq are employed to restrain displacements in the axial, tangential, radial and rotational directions at both ends of the cylindrical shell. The dimensionless internal forces and displacements at both ends satisfy the relationship
~ ¼e ~ x ¼e ~x ¼ e ~x ¼e ~ S ~ V ~; N ~ kq f; kw w; kv v ku u M xq
(12)
where e ks ¼ ks R2 h =K, s ¼ u; v; w; q are dimensionless stiffness coefficients. Classic boundary conditions including S (Simply-support), C (Clamped) and F (Free) are special cases of E (Elastic restrain) boundary conditions. When the dimensionless stiffness coefficients are generally taken as 106, the displacement result at the end tends to be zero which is approximately equivalent to the clamped boundary. The not restrained displacement is Free boundary if the dimensionless stiffness coefficient is set as 0. By identifying the line numbers of the determinate state vectors in Zðx1 Þ, ZðxN Þ, deleting the corresponding row and column in the coefficient matrix in Eq. (11) and expanding the coefficient matrix determinant, the dispersion equation of the cylindrical shell can be
Please cite this article as: Wang, X et al., Vibro-acoustic modelling of immersed cylindrical shells with variable thickness, International Journal of Naval Architecture and Ocean Engineering, https://doi.org/10.1016/j.ijnaoe.2019.12.003
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Fig. 6. Variations of natural frequencies of the cylindrical shell versus the coefficient A.
constructed to gain the natural frequencies of the stiffened cylindrical shell.
2.6. The acoustic responses A virtual interior surface S0 is constructed with virtual sources by shrinking the real surface S. The fluid load p is expressed as:
pðPÞ ¼ ∬ S0 sðOÞKðP; OÞdS0
(13)
respectively. sðOÞ denotes the unknown distribution function of the virtual source strength density. The source strength function K(P,O) and the distribution function sðOÞ can be expanded into a complex Fourier series. The distance d(P,O) equals qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 r P þ r O 2rP rO cosðqP qO Þ þ ðzP zO Þ . A complex distance d satisfying ð1 þigÞd is employed to avoid the non-uniqueness solution at certain characteristic frequencies. The integral interval ð0 L0 Þ and the integral interval ð0 2pÞ are divided into M and N equal divisions respectively, Eq. (13) yields
where P and O denote the external field point and source point,
Fig. 7. Comparison of sound pressure level of Model Ⅳ.
Fig. 8. Comparison of sound pressure level of Model Ⅴ.
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3. Results
pðPÞ ¼
þ∞ X
þ∞ X
~ inqP cmn Kmn ðPÞe
(14)
n¼∞ m¼∞
~ ¼ 2pL0 PM1 PN1 K½P; ~ k ; k rL eimk1 2Mp eink2 2Np is the where Kmn ðPÞ 1 2 O k1 ¼0 k2 ¼0 MN generalized sound pressure load. L0 denotes the meridian line ~ can be obtained by the length of the virtual source surface. Kmn ðPÞ discrete Fourier transform method quickly and accurately. By expanding the potential function into a complex Fourier series, the normal derivative of fluid load p is expressed as þ∞ þ∞ X X vpðPÞ 0 ~ inqP ~ OÞe ¼ cmn Kmn ðP; vnP n¼∞ m¼∞
(15)
The whole points (xi , i ¼ 1, 2, …, m) on the meridian line of real surface S should satisfy the Neumann boundary condition, which yields
K 0n c n ¼ r0 u2 wn ; n2ð∞ þ ∞Þ
(16)
where wn ¼ f wn ðx1 Þ / wn ðxm1 Þ wn ðxm Þ gT and cn ¼
T cm;n / cm1;n cmn . r0 denotes the density of the fluid media. u denotes the circular frequency. The radial displacement w satisfying the linear superposition principle, one can get
~ ¼ wf ðPÞ ~ þ wn ðPÞ n
þ∞ X
3.1. Convergence analysis The convergence of numerical results involves the intercepting mode error. The geometric parameters of a simply-supported cylindrical shell are given as follows: radius R ¼ 0.2 m, length L ¼ 0.6 m, thickness h ¼ 3 mm, size of the internal stiffener is 2 mm 30 mm, stiffener interval is 0.2 m. The radial unit force acts on the position (L/20o , 0.2 m) (cylindrical coordinate) of the internal surface. The measuring point is located at the position (L/20o , 1.2 m). The frequency domain is 40~4 kHz and the frequency step is 40 Hz. The shell and stiffener are made of steel, with density r ¼ 7850 kg/m3, Poisson's ratio m ¼ 0.3, Young's modulus of elasticity E ¼ 2.06 1011 Pa, and damping loss factor h ¼ 0.01. The sound speed of the fluid c0 ¼ 1500 m/s and the density r0 ¼ 1000 kg/m3. As shown in Fig. 2, a convergence is brought about with few modes in low frequency range. More modes are employed to ensure high accuracy as the frequency increases. The number of axial wavenumber m and circumferential wavenumber n take 0e5 and 1e35 respectively in order to realize the accuracy guarantee in the calculated frequency range.
3.2. The free vibration a) Comparison: Model I
p ~ cmn wmn ðPÞ
(17)
m¼∞
~ denotes the radial displacement under the external where wfn ðPÞ load FðxÞ. wpmn ðxÞ denotes the radial displacement under generalized sound pressure load Kmn . ~ on the meridian line of the surface S After collocating points P and substituting Eq. (17) into Eq. (16), one can get
U n c n ¼ Q n ; n2ð∞ þ ∞Þ
(18)
0 ðPÞ ~ r u2 wp ðPÞ, ~ ~ where U q2mþ1 ¼ Kmn Q q1 ¼ r0 u2 wfn ðPÞ. mn 0 Moore-Penrose generalized inverse method is employed to solve the unknown coefficient vector c n. Then the radiated noise of immersed cylindrical shells is obtained by the pluralized wave superposition method (PWSM).
In order to prove the effectiveness of the present method, the geometric and physical parameters of the cylindrical shell (Model I) with simply supported boundary conditions are given as follows: radius R ¼ 0.1 m, length L ¼ 0.2 m, thickness h ¼ 2 mm, density r ¼ 7850 kg/m3, Poisson's ratio m ¼ 0.3 and Young's modulus of elasticity E ¼ 2.1 1011 Pa. The natural frequencies for each order (m, n) are calculated and compared with the exact solution as shown in Table 1. The natural frequencies obtained by the present method are basically consistent with the exact solution. The maximum relative errors are no more than 0.5%, which demonstrates the reliability of the present method on free vibration of the cylindrical shell. b) Comparison: Model II The emphasis of this part is to fully verify the effectiveness of the present method on natural frequencies of the submerged cylindrical shell. The geometric and physical parameters of the model II (Zhang, 2002) are given as follows: radius R ¼ 1.0 m, length L ¼ 20 m, thickness h ¼ 10 mm, Poisson's ratio m ¼ 0.3, Young's modulus of elasticity E ¼ 2.1 1011 Pa and density r ¼ 7850 kg/m3. Sound speed c0 in water equals 1500 m/s. The density of water r0 ¼ 1000 kg/m3. For a clamped cylindrical shell, the natural frequencies of the immersed shell are compared with analytical results and FEM result, as shown in Table 2. The maximum relative error is no more than 3%, which shows that the present method is consistent with FEM results and wave propagation method. It also demonstrates the reliability of the present method on free vibration of the immersed cylindrical shell. c) Comparison: Model III
Fig. 9. Comparison of sound pressure level of Model Ⅵ.
Model III is constructed to validate the reliability of the method on free vibration of the cylindrical shell with variable thickness. The geometric and physical parameters of the Model III are given as follows: radius R ¼ 0.1 m, length L ¼ 0.2 m, Poisson's ratio m ¼ 0.3,
Please cite this article as: Wang, X et al., Vibro-acoustic modelling of immersed cylindrical shells with variable thickness, International Journal of Naval Architecture and Ocean Engineering, https://doi.org/10.1016/j.ijnaoe.2019.12.003
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Young's modulus of elasticity E ¼ 2.1 1011 Pa and density
r ¼ 7800 kg/m3. The thickness h equals h0 ð1 þ Ax=LÞB , which is the function of the axial coordinate value x. Both A and B are suggested to be 1. h0 is set to 2 mm. The natural frequencies of the simply supported cylindrical shell with variable thickness are compared with FEM results, as shown in Table 3. From Table 3, the natural frequencies obtained by the present method are in good agreement with FEM results. The maximum relative difference is no more than 3%. To keep both the thickness of both ends to be constant (the thickness of left end hl ¼ 2 mm and the thickness of right end hr ¼ 4 mm), the model III was employed to analyze the influence of thickness coefficient A and B on free vibration of the cylindrical shell with various thickness. When circumferential wavenumber n is 2, 4, 6 and 8 respectively, the relation between natural frequencies and thickness coefficient B was analyzed, as shown in Fig. 3 and Fig. 4. The effect of the thickness coefficient B can be ignored when circumferential wavenumber n is less than 4. The difference becomes more obvious as the circumferential wavenumber n and axial wavenumber m increase. For a given wavenumber n, the curve shows the antisymmetric characteristic. When the thickness coefficient B is negative, the natural frequency goes up as the thickness coefficient B increases. It comes to an opposite conclusion when the thickness coefficient B is positive. When circumferential wavenumber n was 2, 4, 6 and 8 respectively, the relation between natural frequencies and thickness coefficient A was also analyzed, as shown in Fig. 5 and Fig. 6. The effect of the thickness coefficient A is in accordance with that of the thickness coefficient B when circumferential wavenumber n is less than 4. The only difference is that the coefficient A affects the natural frequency when m ¼ 1 and n ¼ 1. The difference of natural frequencies become more obvious as the circumferential wavenumber n and axial wavenumber m increase. The natural frequency goes up as the thickness coefficient A increases when n ¼ 4, 6 and 8. When n ¼ 2 the curve is approximately a horizontal line, which means thickness coefficient A has little impact on the natural frequency.
3.3. The acoustic response a) Comparison analysis: Model Ⅳ
Fig. 10. Comparison of sound pressure level of Model Ⅶ: (a) hydrophone 1, (b) hydrophone 2, (c) hydrophone 3.
The geometric parameters of the Model Ⅳ are given as follows: radius R ¼ 0.175 m, length L ¼ 0.6 m, thickness h ¼ 2 mm. The centre of the left end of the cylindrical shell was defined cylindrical coordinate origin. The radial unit force acts on the point (L/ 2,0 ,0.175) (cylindrical coordinates) of the cylindrical shell with excitation frequency f ¼ 4 kHz. The hydrophone is fixed at the position of 1 m from the external surface of the shell. The physical parameters of the shell are given as follows: Poisson's ratio m ¼ 0.3, Young's modulus of elasticity E ¼ 2.1 1011 Pa, density r ¼ 7850 kg/m3, and loss factor h ¼ 0.01. Sound speed c0 in water equals 1500 m/s. The density of water r0 ¼ 1000 kg/m3. The simply supported boundary conditions at both ends are assumed. The results contrast of sound pressure level Lp is yielded by adopting the present method, experimental method and B. Laulgnet's method respectively, as shown in Fig. 7. The error curve shows the maximum error of the sound pressure level between the present method and experiment results. It can be seen from Fig. 7 that the error of sound pressure level is 0.5%e5% at other angles except at 60 , which accords with the allowable range of error. It proves that the present method is feasible, effective and sufficiently accurate and can be applied to predict sound radiation of cylindrical shells.
Please cite this article as: Wang, X et al., Vibro-acoustic modelling of immersed cylindrical shells with variable thickness, International Journal of Naval Architecture and Ocean Engineering, https://doi.org/10.1016/j.ijnaoe.2019.12.003
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b) Comparison analysis: Model Ⅴ The geometric parameters of the Model Ⅴ with simply supported boundary condition are given as follows: radius R ¼ 0.2 m, length L ¼ 0.6 m, thickness h ¼ 3 mm. Sectional dimensions of the stiffener is 2 mm 30 mm. The stiffeners spacing Dl ¼ 0.2 m. The radial unit force acts on the point (0.3 m, 0 , 0.2 m) (cylindrical coordinates) of the shell with excitation frequency f ¼ 4 kHz. The hydrophone is fixed at the position of 1 m from the external surface of the shell. The physical parameters are the same as model Ⅳ. It can be seen from Fig. 8 that the relative error of sound pressure level between the present method and wave propagation method is no more than 6%. The results from the present method also are consistent with the experimental values, except for some minor differences in some partial angle. Hence, the present method proposed can be applied to vibro-acoustic behaviour of the stiffened cylindrical shell. c) Comparison analysis: Model Ⅵ Model Ⅵ with the variable thickness is constructed to validate the availability and accuracy of the present method on the sound radiation of the cylindrical shell with the variable thickness. The geometric and physical parameters of the Model Ⅵ with clamped boundary condition are given as follows: the variation coefficient A ¼ 1, the power coefficient B ¼ 1. The other geometric and physical parameters are the same as model Ⅴ, except without ring ribs. The radial unit force acts with excitation frequency range 10 Hz-4kHz. The observation point located at position (0.3 m, 0 , 1.2 m), in which acoustic responses of Model Ⅵ are calculated by coupled FEM/BEM. C3D8R elements in software ABAQUS are employed to generate linear hexahedron meshes of the cylindrical shell, the size of which is set to 4 mm. The external flow field radius is taken as 2 m. A hemispherical flow field is meshed by AC3D4 acoustic tetrahedral elements, the size of which is from 0.004 m to 0.0125 m. The results comparison of radiated noises at the observation point between the present method and coupled FEM/BEM is shown in Fig. 9. The results from the present method are consistent with the results from coupled FEM/BEM at 10 Hze2100 Hz. Although the sound pressure level is slightly different at the resonance peak value in the 2100~4 kHz range, the sound pressure values resulted from the present method is in accord with numerical results basically. There are many reasons for the difference. The process of equal divisions along the integral interval in Eq. (14) could cause the error of approximation. Both the substitution of the finite field for infinite fluid field and the setting of Tie connection at the interface also lead to the numerical error in software ABAQUS. The comparison provides the validity and reliability of the present method, which can be applied to sound radiation of immersed cylindrical shells with the variable thickness. d) Comparison analysis: Model Ⅶ The test model of a ring-stiffened cylindrical shell is constructed and prepared to prove the effectiveness of the present method. Model Ⅶ has the following material properties: Length L ¼ 0.8 m, Radius R ¼ 0.3 m. Shell thickness t ¼ 4 mm. Sectional dimensions of the stiffener is 40 mm 4 mm. The stiffeners spacing Dl ¼ 0.16 m. The other physical parameters are the same as the Model Ⅳ. 15 mm thick caps are screwed at the ends of the cylinder. The shaker is fixed on the end cap in order to excite the point (0.4 m,0 ,0.3 m) of Model Ⅶ. There are 3 hydrophones (BK-8104) is placed 1.0 m far from the cylindrical shell, which was set up in the anechoic tank to measure the sound pressure. Comparison of sound radiation curves between model test and the present method with the linear sweep
frequency (100 Hz-2kHz) is given in Fig. 10. It can be observed that although the sound pressure is slightly different at the resonance peak value in the 500e2000 Hz band, the sound pressure result from PTMM basically is in accord with test result. When the internal medium of the experimental model is full of air, the coupling between structural mode and acoustic cavity mode has little effect on the vibro-acoustic response of the cylindrical shell. In general the present method provides enough verification and accuracy, which can be applied to vibro-acoustic behaviours of immersed stiffened cylindrical shells. 4. Conclusions The vibro-acoustic model of the immersed stiffened cylindrical shell with various thickness is developed in this paper to predict the vibro-acoustic performance by combining PTMM and PWSM. Based on thin shell theory, a global matrix formulation as the governing equation is constructed with a set of field matrixes of stiffened cylindrical shell segments with various thickness. Then the dynamic behaviours can be directly solved by PTMM. The fluid load simulated by PWSM is employed to evaluate the radiated noise of the immersed cylindrical shell. Free vibration of three different cylindrical shells including cylindrical shell, immersed cylindrical shell and variable-thickness cylindrical shell are constructed and analyzed to illustrates the feasibility of the present method. Acoustic responses of four different cylindrical shells including cylindrical shell, immersed cylindrical shell, stiffened cylindrical shell and variable-thickness cylindrical shell are also constructed and analyzed to demonstrate its effectiveness. It is worth to point out that the proposed method is conveniently extended to analyze the vibro-acoustic behaviours of other elastic shells like conical shells, double-walled shells and combined shells with various thickness. Declaration of competing interest We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled, “Vibro-acoustic modelling of immersed cylindrical shells with variable thickness”. Acknowledgements The authors would like to thank the anonymous reviewers for their valuable comments. This paper was financially supported by China Scholarship Council (201806955052), National Natural Science Foundation of China (51779201, 51609190) and Natural Sci- Q5 ence Foundation of Hubei Province (2018CFB607). Appendix A
Trði; iÞ ¼ 1; i ¼ 1; 2; 3/8 R Trð5; 1Þ ¼ K
("
EI2 R3b
2
n þ
GJ R3b
# 2
n
h þ eh
" EI2 R4b
4
n þ
GJ R4b
#) n rAu 2
2
Please cite this article as: Wang, X et al., Vibro-acoustic modelling of immersed cylindrical shells with variable thickness, International Journal of Naval Architecture and Ocean Engineering, https://doi.org/10.1016/j.ijnaoe.2019.12.003
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IJNAOE293_proof ■ 30 December 2019 ■ 11/11
X. Wang et al. / International Journal of Naval Architecture and Ocean Engineering xxx (xxxx) xxx
Trð5; 4Þ ¼ # " 8 9 EI2 EI2 2 GJ 2 GJ 2 2 > > > > R en en þ R n r I u p > > 3 b 3 3 3 b > > < = R R R R b b b b h " # > > K> > > > > : þe EI2 en4 þ EI2 Rn2 GJ en2 þ GJ Rn2 rAu2 e > ; R4b R4b R4b R4b " # R2 EI1 Rb 3 EA Rb n þ 2 Trð6; 2Þ ¼ h K R4 R Rb R b Trð6; 3Þ ¼ h
" # R2 EI1 Rb 4 EA e 2 2 n n þ r A u 1 þ R K R4 R R2b b
Trð7; 2Þ ¼ h
" # R2 EI1 Rb 2 EA Rb 2 2 Rb n n þ r A u K R4 R R R2b R b
Trð7; 3Þ ¼ h
# " R2 EI1 Rb 3 EA R e 2 e n þ n r A u n R R K R4 R R2b b
Trð8; 1Þ ¼ h
# " R2 EI2 4 GJ 2 2 n þ n r A u K R4 R4b b
# " h R2 EI2 4 EI2 2 GJ 2 GJ 2 2 Trð8; 4Þ ¼ 4 en þ 4 R1 n 4 en þ 4 R1 n rAu e R K Rb Rb Rb Rb where Rb ¼ R þ e e is the offset of the rib. E, G are the Modulus of elasticity and shear of the rib. r is the density of the rib. A is the sectional area of the rib. I1 , I2 and J are the moment of inertia in xdirection, y-direction and polar moment of inertia. Q6
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Please cite this article as: Wang, X et al., Vibro-acoustic modelling of immersed cylindrical shells with variable thickness, International Journal of Naval Architecture and Ocean Engineering, https://doi.org/10.1016/j.ijnaoe.2019.12.003
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